A modified fractional short circuit current MPPT and multicellular converter for improving power quality and efficiency in PV chain

This article presents the contribution of multicellular converters in improving of the quality of power produced in photovoltaic chain, with the aim of exploiting the maximum power produced by the photovoltaic generator with low oscillations around of the maximum power point (MPP) at steady state and to reduce switching losses. After modeling the multicellular parallel boost converter, fractional short circuit current (FSCC) MPPT was modified to get an estimated photocurrent as a reference to control the inductance current for good functioning of the converter in pursuit of the maximum power point. To verify the performance of the proposed solution, the system was submitted to irradiance and temperature variations. The simulations carried out in the Matlab/Simulink environment presented satisfactory results of the proposed solution, in comparison with the high-gain quadratic boost converter we have a response time of 0.04 s, power oscillations at maximum point around 0.05 W and efficiency of 99.08%; in comparison with the interleaved high-gain boost converter the results show a response time of 0.1 s for the transferred power, a very low output voltage ripples of 0.001% and 98.37% as efficiency of the chain. The proposed solution can be connected to a grid with a reduction of level of the inverter and active filter.


Introduction
Electrical energy is an essential element for development and improvement of living conditions of a society; its shortage disrupts household life and leads to a slowdown in activities in certain areas of country.The promotion of the use of renewable energies is among the initiatives undertaken by several countries to alleviate electrical energy problems while preserving the environment.Several research projects are being carried out to develop the field of renewable energies.Among renewable energy sources, photovoltaic energy presents itself as the most promising starting from its raw material.The design of optimized photovoltaic systems is by nature difficult due to the fact: on the source side, the power produced by a photovoltaic generator strongly depends on environmental conditions (irradiance and temperature) [1][2][3][4] but also the overall state (age) of the system; on the load side, which the nature can be continuous or alternative, each has its own behavior which can be random [1,2,5,6].For optimal operation of the photovoltaic generator (GPV), the introduction of an optimal power point tracker is necessary.The power point tracker built around a static converter acting as a source-load adapter with the aim of forcing the generator to operate at its maximum power point (MPP).For the control of the static converter, several maximum point tracking techniques forcing the photovoltaic generator to follow the optimal point despite the environmental conditions have been developed.[7][8][9][10][11][12].However, the maximum power point varies depending on the surface of the photovoltaic generator, the temperature and the irradiance [13][14][15][16]; the aim of the control technique is to automatically modify the duty cycle to bring the generator to its maximum operating point whatever the weather conditions or load variations that may occur as illustrated on Fig 1 [17].Several maximum power point tracking algorithms have been developed , according classification made by [29] the different technics can be classified in three large groups: Indirect techniques (fraction of V OC and fraction of I SC . ..), direct techniques (perturb and observe (P&O), incremental conductance. ..) and techniques by intelligent control (fuzzy logic, sliding mode, etc.).Good dynamic behavior is very useful in the event of rapid variation of the source (irradiance) or the characteristics of the load.
The static converter playing the role of interface between the source and the load most used in photovoltaic applications is the boost converter, the basic boost converters although they raise up the voltage at their output, are more suitable for low power applications [45][46][47][48][49].For high power applications, several converters structures have been used: fly-back converters, buck-boost converters, multi-cell converters, high-gain quadratic boost converters.
Fly-back converters [49][50][51][52] first perform a DC-AC conversion to obtain a high amplitude alternative voltage, then an AC-DC conversion; buck-boost converters [53,54] presented high dynamic losses and low efficiency.Multicellular converters appeared in the 1990s, offering the possibility of reducing voltage or current stresses in power switches.Parallelizing the base converters allows the use of low current levels per block to promote long component life and increase the reliability of the base converters [48,55].Still contributing in high power applications, a single switch quadratic power converter based on a switched capacitor to achieve high voltage gains and low voltage constraints on power components and reduce the complexity of controller designs has presented by [43,49] proposes a high-gain quadratic boost converter (HG-QBC) to overcome the limitations of conventional boost converter and [44] presents a solar-powered interleaved high-gain boost converter (IHGBC) that increases the voltage gain with less ripples in the output voltage compared to existing DC-DC converters.
The massive use of non-linear loads designed using power electronics converters degrades the quality of electrical energy by generating harmonic currents; these power converters can also exhibit chaotic effects or behavior at certain switching frequencies [56,57].According to [58], increasing the number of switching cells connected in parallel would lead to a reduction in output current ripples (Fig 2).Non-perfect coupling between the photovoltaic generator and the load, oscillations around the maximum point in steady state and occasional loss of tracking the maximum point during rapid change in climatic conditions, and chaotic behavior in static converters are all among the problems which degrade the energy efficiency of a photovoltaic chain.
In this article, the objective is to contribute for improving of energy efficiency and power quality of photovoltaic system by optimizing the power produced based on the operating of GPV on principle of current controlled voltage source (CCVS) and the use of multicellular converters; firstly to improve the indirect short-circuit current fraction (FSCC) method by estimating a photo-current (reference current) to control the inductance current in order to follow the maximum power point (MPP), secondly use the parallel multicellular converter integrating synchronous rectification to reduce switching losses in the switches (by reducing the switching frequency), operating in interleaving mode for the reduction of power oscillations in steady state.
This paper is structured as follow: section 1 presents the introduction where the state of art is presented, section 2 reserved for models, control techniques and calculation of efficiency, in section 3 is presented the simulation results and discussions, finally conclusion in section 4. qq2.Method

Photovoltaic generator model
In its constitution, photovoltaic generator (GPV) contains a set of elementary photovoltaic cells; to obtain the desired electrical characteristics (short-circuit current I SC , open circuit voltage V OC ), the elementary cells are connected in series and/or parallel.A photovoltaic cell can be illustrated by its equivalent following diagram [59,60]: By applying KCL (Kirchhoff's current law) on node N in Fig 3: with: By replacing I D et I R by their expressions in Eq (1) we obtain: Where I is the cell current (A), I ph is the photocurrent (A), V is the cell voltage (V), R S is series resistance of the cell (Ω), R P is parallel resistance of the cell (Ω), T is the cell temperature (K), q is electron charge (q = 1,6.10−19 C), I O the saturation current (A), K is Boltzmann constant (k = 1,3854.10−23 J/K), n is the diode quality factor.The current in the cell is maximum during the short circuit (V = 0,I = I SC ), Eq (4) becomes: In the ideal case (R S �0,R P �1) the short-circuit current becomes:

Modeling of boost converter
The converter is modeled after analysis of the different operating sequences of the switches, the durations of which are fixed by the command [61].The differential equations system given by Eq (7), represents the state equation of the converter.
The state equation of the boost converter being non-linear, its affine form can be written: with: X = [I L V S ] T The state vector; U 1 ,U 2 : Discontinues command; The state matrix f ¼ According to [62], the system is controllable 2.2.2 Multicellular boost converter.In reality, static converters can only provide a chopped voltage (or current), due to the qualification of the power electronics as forced or natural switching electronics [57].To reduce the undesirable effects of the output voltage chopping, and thus move a little more towards the "ideal converter", solutions such as increasing the number of levels available at the output of static converter, increasing the switching frequency of the output voltage so as to push the switching harmonics further and optimization of the control strategy so as to ensure the best possible tracking of the reference signal had been adopted.However, putting several switching cells in parallel presents itself as a better solution for reducing the undesirable effects of the output current chopping (Fig 2 ), this through to the magnetic coupler which acts as a filter by only allows pass the current that harmonics content is multiple of the cells number connected in parallel (S3 Fig) [63,64], Fig 5 shows the cases where 3 and 5 cells are connected in parallel.
The state equation of the converter made up of P cells put in parallel described in Fig 6 is given by the differential equation system described by Eq (10).For operation in multi-phase or interleaving mode, the control signals must be shifted from each other by Delay ¼ period P , the current in each branch must be value I branch ¼ 1 with a n = 1−S cn , Its affine form is: with: f: state matrix; X = [I L1 I L2 . ..I LP V S ] T The state vector; U 1 ,U 2 ,. ..,UP ,U P+1 : discontinues command.For 5 cells connected in parallel, the state equation is given by Eq (12): ; ; Controllability: det[g 1 g 2 g 3 g 4 g 5 g 6 ]6 ¼0, then the system is controllable.According to Eqs ( 9) and ( 14), the system can be control from the current flowing through the inductance.

Method based on control of the inductance current
In steady state or for small variations V pv Voltage, Eq (15) becomes: From Eqs ( 17) and ( 6), we will choose the reference current by estimating the photocurrent due to the high influence of irradiance on the output current from GPV.The relationship linking the photocurrent to the irradiance and temperature is given by [61,65]: Where α is the short circuit current temperature coefficient (A/K), T C is the temperature of the cell (K), T r is the reference temperature (25˚C or 298K), G is the irradiance (W/m 2 ) and G r the reference irradiance (1000/m 2 ).
When T C = T r , According to [66] for a polycrystalline silicon material, α = 0,0021A/K; according to [67] for a silicon material, α = 0,00238A/K for polycrystalline structure and 0,00175A/K for the single crystal structure.Given these values, Eq (18) can be expressed: In control technics based on the proportionality relation of the linear relation in first approach between I OPT and I SC described by [68,69], Eq (22) in Eq (20) give the following relation: K I being a current factor generally between 0,78 et 0,92; let's put 1 K I � 1, then Eq (23) can be written: The reference current can be formula in the form 2.3.2Controller.To ensure control of current through the inductance, the controller whose goal is to oblige the inductance current to follow the reference current by automatically updating the corresponding duty cycle is built around PI regulator whose the transfer function is given by Eq (25): From Fig 7, The parameters K P and K I must be chosen so as to make if possible ε(s)�0˚; for this the particle swarm optimization (PSO) algorithm which the objective function shown on Fig 8 and describe by Eq (28) was used for choose the values of K P and K I .
Where, V i+1 ,x i+1 are respectively the updated speed and position of each particle i; V i ,x i are respectively the actual speed and position of particle i; μ 1 ,μ 2 ,μ 3 2 [0,1]; x ip ,x g are respectively the best position visited by the particle i and the best position visited by the swarm.

Method of control by sliding mode
The diagram of control by sliding mode illustrated in Fig 9 , shows that the reference current generated previously can be used as reference current in sliding mode control.The Lyapunov criterion is used to define switching functions because of the simplicity of its implementation.
The quadratic Lyapunov function is define around energies stored in capacitors and inductances as follow: Where H is a diagonal constant matrix containing inductive and capacitive elements; According to Eqs ( 9) and ( 14), we pose the error ΔX = X ref −X as follow: . . .The system will be stable in a closed loop if the derivative of the Lyapunov function is negative, Switching functions are defined through the following relationship: For P = 5, S i ¼ À DX T :H:g i ; i ¼ 1; 2; 3; 4; 5; 6 Error ΔX is stable if S i = 0˚; for S 6 = 0, we obtain: Thus giving the current to pass through each branch for multicellular use.

Calculation of efficiency
According to [17], the calculation of the efficiency in a photovoltaic chain is product of the MPPT efficiency (η MPPT ) and the conversion efficiency (η CONV ) of the static converter.The MPPT efficiency determines the effectiveness of control technics in terms for tracking the maximum power point; it is given by the following relation: Where P pv is the power delivered by the photovoltaic generator, P MPP the maximum power of the photovoltaic generator at the maximum power point.
The efficiency of a static converter can be defined as its ability to transfer at its output the maximum of available power at its input [49]: P pv is the power delivered by the photovoltaic generator which becomes the input power of converter, P out is the power transferred to the converter output.
The total efficiency of the chain is given by:

Simulation results and discussion
In this part, we present the different simulations carried out in Matlab/Simulink environment and their results.We compared the suggested solutions based on the control of the inductance current and multicellular firstly to some traditional MPPT technics control, and to some solutions proposed by others works in term of improvement of power quality.Simulations were carried out in various cases: under constant irradiance, variable irradiance and variable temperature.The photovoltaic module used is type Kyocera Solar KC200GT which some characteristics depending on the irradiance under a constant temperature of 25˚C are given in S1  1 give the designed parameters for simulation.

functioning of conventional boost converter with the suggested MPPT for various duty cycle
The proposed MPPT was simulated on the conventional boost to check if it responds to operating criteria of this converter.Some duty cycles were used: 0.  To make the comparison between the different maximum point search techniques in terms of speed, stability, precision, an analysis was carried out and grouped in Table 2, and in Table 1.Designed parameters for simulation.

Components
Values Switching frequency (F sw ) 5KHz

Comparison of proposed solution with HG-QBC
The proposed solution is compared to the high-gain quadratic boost converter (HG-QBC) proposed by [43] in order to appreciate its performances.The An analysis is made between the performances of the proposed solution and the performances of the HG-QBC proposed by [43] and is presented in Tables 4  and 5.By analyzing values presented on Table 5 to values on S1

Comparison of proposed solution with IHGBC
Among the solutions for improving the performance of boost converters dedicated to PV applications, there is the interleaved high-gain boost converter (IHGBC) proposed by [44].This converter has the capacity to increase the voltage gain with less ripple output.6 presents the comparison made between the proposed solution and the IHGBC.In spite of slow response time of proposed solution facing the IHGBC, Table 6 show best performances of the proposed solution in term of ripples minimization and maximum transferred power to output.

Conclusion
In this paper, the objective was to use the principle of current controlled voltage source and the parallel multicellular converter to improve the energy efficiency of a photovoltaic chain.A method by modification of the technics based on fraction of the short circuit current (FSCC) with the aim of generating a reference current to control the inductance current in order to control the static converter to follow the MPP has been proposed, this reference current was used as reference current in sliding mode control.The parallel multicellular converter integrating synchronous rectification through the advantages offered by the interleaving mode has  been used to reduce power oscillations and switching losses in switches.The simulation results obtained show the effectiveness of the method by estimating the photocurrent as a reference current used for the pursuit of MPP, responds effectively whatever the environmental conditions, and the contribution of the multicellular converter in reducing oscillations of power around the MPP in steady state and the losses in the switches.The results show a response time of 0.04 s, power oscillations at maximum point around 0.05 W and efficiency of 99.08% facing the high-gain quadratic boost converter.Then by comparing with the interleaved highgain boost converter the results show a response time of 0.1 s for the transferred power, a very low output voltage ripples of 0.001% and 98.37% as efficiency of the chain.Multicellular converter integrating synchronous rectification can be used in many power system, the control technique based on the control of inductance current is suitable for static converter which the inductance current should be control for its good functioning.The proposed solution can be connected to a grid by reducing the level of the inverter and active filter.

2 . 2 . 1
Single cell boost converter.Fig 4 represents the structure of the single-cell Boost converter where T is a switch controlled by the signal S C .

2. 3 . 1
Principle of the method.The diagram of the method is illustrated in Fig 7.
Fig 7 gives according to KCL, the relationship linking the PV generator current (I pv ), inductance current (I L ) and the capacitor current (I C ):

25 (
Fig 10), 0.5(Fig 11)  and 0.75 (Fig12).for 0.5 duty cycle value, the output voltage of the boost converter must be equal to twice the value of its input voltage, this is verified byFig 11.

3. 2
Comparison with some MPPT technics under constant irradiance G = 1000W/m 2 and variable irradiance for temperature T = 25˚C The proposed solutions are compared to some MPPT technics such as: perturb and observe (P&O) in Fig 13, fraction short circuit (FSCC) in Fig 14, fuzzy logic (Fig 15) and hybrid sliding mode assisted by P&O in Fig 16.The operations were simulated under the conditions of constant irradiance, variable irradiance according to the profile of Fig 17, both under a constant temperature of 25˚C.

3. 3
Simulation of the proposed solution under temperature variations at constant irradiance G = 1000W/m 2 The proposed solution was subjected to a temperature variation that the profile is illustrated on Fig 20, we notice in Fig 21A a very weak influence of the temperature on the current, according to Fig 22B we observe the tracking of the maximum point of the power of the GPV with a response time of 0.002 seconds and a response time of 0.02 seconds for the transferred power.These results agree with the values described in S2 Table and S2 Fig.

Fig 14 .Fig 15 .
Fig 14.Powers curves with conventional boost based on FSCC MPPT.(a)Constant irradiance.(b)Variable irradiance.https://doi.org/10.1371/journal.pone.0309460.g014 Figs 23-26 show that the proposed solution can operate in a wide voltage range.Fig 26B show more the behaviors facing the different change of variations according to profile presented on Fig 23A.

Fig 29 .Fig 30 .
Fig 29.(a) Currents and (b) voltages under variable irradiance.https://doi.org/10.1371/journal.pone.0309460.g029 Table and S1 Fig for different irradiance values, we observe that the proposed solution presents best results in term of precision of tracking.

Table 2 . Comparison among some solutions in terms of powers, response times and oscillations.
MPP is maximum power of GPV at MPP, P pv is power delivered at out of GPV, T r_PV is response time.

Table 3 . Comparison among some solutions in terms of transferred powers and efficiencies.
Where P out is output power of the converter, η MPPT is MPPT efficiency, η CONV is conversion efficiency, η CHAIN is the total efficiency of the chain. https://doi.org/10.1371/journal.pone.0309460.t003

Table 5 . Performance analysis of the proposed solution and the HG-QBC under irradiance variation.
Where V out ,I out ,P out are respectively output voltage, output current and output power of the converter. https://doi.org/10.1371/journal.pone.0309460.t005